$$\text{Let } f(x) = \left\{ \begin{array}{lr} 1 & : 0 \leq x\leq1 \\ 2 & : 1 < x \leq e\\ 3 & : e < x \leq \pi \\ 4 & : \pi < x \leq 4 \end{array} \right.$$
How can we verify Riemann's condition for $f$ and is it possible, given $\varepsilon > 0$, to find an explicit $\delta >0$ so that every partition $\mathcal{P}$ with mesh$(\mathcal{P}) < \delta$ satisfies Riemann's condition for $\varepsilon$.
Suppose $\delta<\pi-e$ and $P$ has mesh $\delta$. Then there are at most $3$ subintervals of $P$ on which the supremum and infimum of $f$ are not equal (since there are $3$ jumps, each of which is separated by at least $\pi-e$, and $f$ is piecewise constant). The supremum and infimum will differ by $1$ (because all the jumps are of size at most $1$, and each subinterval will only overlap at most one of the jumps, since $\delta<\pi-e$). Hence the difference between the upper and lower sum for $P$ will be at most $3\delta$.
This argument, along with the standard argument that Darboux's condition implies Riemann's condition, tells us that Riemann's condition for $\varepsilon$ is satisfied provided $\delta<\min \{\pi-e,\varepsilon/3 \}$.
Note that part of this argument is that $\min \{ 1,e-1,\pi-e,4-\pi \}=\pi - e$. I'll leave it to you to prove that.
For example, if we consider the upper sum with $\varepsilon=0.3$, we get $\delta=0.1$ from the above. Using the uniform partition with $\delta=0.1$ and recalling that $2.7<e<2.8$ and $3.1<\pi<3.2$, we get that the integral is between $8.9$ and $9.2$.